Last digit

What is the last digit of: 3^2046?

$\begin{array}{|rcll|} \hline && 3^{\varphi(10) } \equiv 1 \pmod{10} \quad | \quad \varphi(10) = 10\cdot(1-\frac12)\cdot(1-\frac15) = 4 \\ && 3^4 \equiv 1 \pmod{10} \\\\ && 3^{2046} \pmod{10} \\ &\equiv & 3^{4\cdot 511 +2} \pmod{10} \\ &\equiv & 3^{4\cdot 511}\cdot 3^2 \pmod{10} \\ &\equiv & (3^4)^{511}\cdot 3^2 \pmod{10} \\ &\equiv & (1)^{511}\cdot 3^2 \pmod{10} \\ &\equiv & 3^2 \pmod{10} \\ &\equiv & 9 \pmod{10} \\ &\equiv & 9 \\ \hline \end{array}$

The last digit is 9

Here are the last 10 or so digits........... 2899855929.

3^1  = 3

3^2  = 9

3^3  = 27

3^4  =81

3^5  = 243

3^6  = 729

3^7 =2187 

3^8 = 6561

3^9 =19683

3^10=59049

  You might be able to see a pattern of last digits happening here?  (repeating every four?)

2046 mod 4 = 2           The second one above is 9    I expect 3^2046   ends in 9.